In this paper, we study the problem of learning the weights of a deep convolutional neural network. We consider a network where convolutions are carried out over non-overlapping patches. We develop an algorithm for simultaneously learning all the kernels from the training data. Our approach dubbed deep tensor decomposition (DeepTD) is based on a low-rank tensor decomposition. We theoretically investigate DeepTD under a realizable model for the training data where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to planted convolutional kernels. We show that DeepTD is sample efficient and provably works as soon as the sample size exceeds the total number of convolutional weights in the network.
- Publication Date:
- NSF-PAR ID:
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- Information and Inference: A Journal of the IMA
- Oxford University Press
- Sponsoring Org:
- National Science Foundation
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In this paper we study the problem of learning the weights of a deep convolutional neural network. We consider a network where convolutions are carried out over non-overlapping patches with a single kernel in each layer. We develop an algorithm for simultaneously learning all the kernels from the training data. Our approach dubbed Deep Tensor Decomposition (DeepTD1 ) is based on a rank-1 tensor decomposition. We theoretically investigate DeepTD under a realizable model for the training data where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to planted convolutional kernels. We show that DeepTD is data-efficient and provably works as soon as the sample size exceeds the total number of convolutional weights in the network. We carry out a variety of numerical experiments to investigate the effectiveness of DeepTD and verify our theoretical findings.
Scan‐specific robust artificial‐neural‐networks for k‐space interpolation (RAKI) reconstruction: Database‐free deep learning for fast imaging
To develop an improved k‐space reconstruction method using scan‐specific deep learning that is trained on autocalibration signal (ACS) data.
Robust artificial‐neural‐networks for k‐space interpolation (RAKI) reconstruction trains convolutional neural networks on ACS data. This enables nonlinear estimation of missing k‐space lines from acquired k‐space data with improved noise resilience, as opposed to conventional linear k‐space interpolation‐based methods, such as GRAPPA, which are based on linear convolutional kernels.
The training algorithm is implemented using a mean square error loss function over the target points in the ACS region, using a gradient descent algorithm. The neural network contains 3 layers of convolutional operators, with 2 of these including nonlinear activation functions. The noise performance and reconstruction quality of the RAKI method was compared with GRAPPA in phantom, as well as in neurological and cardiac in vivo data sets.
Phantom imaging shows that the proposed RAKI method outperforms GRAPPA at high (≥4) acceleration rates, both visually and quantitatively. Quantitative cardiac imaging shows improved noise resilience at high acceleration rates (rate 4:23% and rate 5:48%) over GRAPPA. The same trend of improved noise resilience is also observed in high‐resolution brain imaging at high acceleration rates.
The RAKI method offers a training database‐free deep learningmore »
Normalization techniques have become a basic component in modern convolutional neural networks (ConvNets). In particular, many recent works demonstrate that promoting the orthogonality of the weights helps train deep models and improve robustness. For ConvNets, most existing methods are based on penalizing or normalizing weight matrices derived from concatenating or flattening the convolutional kernels. These methods often destroy or ignore the benign convolutional structure of the kernels; therefore, they are often expensive or impractical for deep ConvNets. In contrast, we introduce a simple and efficient Convolutional Normalization'' (ConvNorm) method that can fully exploit the convolutional structure in the Fourier domain and serve as a simple plug-and-play module to be conveniently incorporated into any ConvNets. Our method is inspired by recent work on preconditioning methods for convolutional sparse coding and can effectively promote each layer's channel-wise isometry. Furthermore, we show that our ConvNorm can reduce the layerwise spectral norm of the weight matrices and hence improve the Lipschitzness of the network, leading to easier training and improved robustness for deep ConvNets. Applied to classification under noise corruptions and generative adversarial network (GAN), we show that the ConvNorm improves the robustness of common ConvNets such as ResNet and the performance of GAN.more »
Batch Normalization (BN) is essential to effectively train state-of-the-art deep Convolutional Neural Networks (CNN). It normalizes the layer outputs during training using the statistics of each mini-batch. BN accelerates training procedure by allowing to safely utilize large learning rates and alleviates the need for careful initialization of the parameters. In this work, we study BN from the viewpoint of Fisher kernels that arise from generative probability models. We show that assuming samples within a mini-batch are from the same probability density function, then BN is identical to the Fisher vector of a Gaussian distribution. That means batch normalizing transform can be explained in terms of kernels that naturally emerge from the probability density function that models the generative process of the underlying data distribution. Consequently, it promises higher discrimination power for the batch-normalized mini-batch. However, given the rectifying non-linearities employed in CNN architectures, distribution of the layer outputs show an asymmetric characteristic. Therefore, in order for BN to fully benefit from the aforementioned properties, we propose approximating underlying data distribution not with one, but a mixture of Gaussian densities. Deriving Fisher vector for a Gaussian Mixture Model (GMM), reveals that batch normalization can be improved by independently normalizing with respectmore »
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